Abstract
We derive the nonlinear sigma model as a peculiar dimensional reduction of Yang-Mills theory. In this framework, pions are reformulated as higher-dimensional gluons arranged in a kinematic configuration that only probes cubic interactions. This procedure yields a purely cubic action for the nonlinear sigma model that exhibits a symmetry enforcing color-kinematics duality. Remarkably, the associated kinematic algebra originates directly from the Poincaré algebra in higher dimensions. Applying the same construction to gravity yields a new quartic action for Born-Infeld theory and, applied once more, a cubic action for the special Galileon theory. Since the nonlinear sigma model and special Galileon are subtly encoded in the cubic sectors of Yang-Mills theory and gravity, respectively, their double copy relationship is automatic.
Highlights
As our dimensional reduction effectively projects out all quartic interactions in the NLSM, pion scattering originates entirely from the cubic topologies of gluon scattering
We have proposed a variation of dimensional reduction that excises the NLSM from YM theory as well as BI theory and the SG theory from the extended graviton action
These relations reveal the origin of the kinematic algebra of the NLSM as the higher-dimensional Poincare invariance of an underlying YM theory
Summary
Consider a tree-level color-ordered scattering amplitude in YM theory. As proven in ref. [1], gluons can be transmuted into pions via a simple differential operation,. (2.1) and (2.2) apply to amplitudes in any representation, provided they are written as a function of kinematic invariants in general spacetime dimension. This is possible because the transmutation operators are precisely engineered to be invariant under reshuffling of terms via total momentum conservation and on-shell conditions [1]. (2.1) and (2.2) are manifestly cyclic and permutation invariant, respectively, while the left-hand sides are not. This feature is generic: while transmutation selects two special legs, chosen here to be 1 and n, the final answer is independent of this choice. Combined with eq (2.2), eq (2.3) shows that applying the transmutation twice to an extended graviton amplitude leads to that of SG
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